Abstract
Dual numbers were applied to the dynamics of a rigid-flexible combination system (RFCS) with time-varying configuration in this paper. The six-dimensional spinor form of the motion of flexible modules, including the dual vector, dual momentum, dual inertia operator, dual coupling coefficient operator, and dual-modal coordinates, was derived using the dual numbers that could represent spiral motion in a compact form. On this basis, the integrated dynamic model of a rigid-flexible combination system with a time-varying configuration was proposed. And then, the relative dynamics equations between two rigid-flexible combination systems which both have time-varying configuration were provided. An on-orbit assembly mission of flexible modules transported and operated by free-flying space robots (FFSRs) is presented as an exemplary application of relative dynamics. Simulation results illustrate the complex coupling effects on the relative motion between two rigid-flex combination systems with time-varying configuration.
1. Introduction
FFSRs present great potential for on-orbit assembly missions [1–6]. The implementation of in-space assembly using FFSR can be divided into three stages. The first step is module capture [7], which means that FFSR captures the module through the capture device; the module and FFSR constitute a combination system. The second step is module transportation [8]. The module is transported to the desired position and integrated through combination systems with large translational and rotational maneuvers. The last step is module docking [9]. The adjacent modules are ultimately connected together through docking mechanisms, achieving structural growth. In this paper, the dynamics of module transportation using FFSR are investigated.
Since the initial motion state of the module, including the attitude and position, is unknown during the capture process, one of the key mission requirements in the module transportation process is to adjust the postcapture configuration of the combination system. The aim of the above configuration adjustment is to prepare the best pose for assembly. Several dynamic modeling methods of the operation of free-floating rigid bodies have been proposed. Liu et al. [10] derived the capture and postcapture dynamics equations for the operation of a noncooperative space target by using FFSR. In this model, the influence of the contact force between the FFSR and the assembly module on the whole system motion was considered while catching a floating rigid body. Cyril and Chau et al. [11, 12] have investigated the dynamics for catching and operating a rigid body by assembly robots. In Refs. [13–16], the attitude dynamics of the combined systems consisting of a rigid service robot, a rigid target, and two rigid space manipulators were investigated. Li et al. [17] improved the attitude dynamics of the combined systems with flexible manipulators by considering the vibration influence on the motion of the whole system. The solutions obtained by the dynamics model mentioned above can handle a large number of motion description problems in space-capturing and postcapturing missions.
Another mission requirement of the module transportation process is to quickly transport the modules to the desired positions for integration by using FFSR. The foundation for meeting this requirement is to establish an accurate and engineering-applicable relative dynamics model for assembly robots. A lot of work has been done to describe the relative motion between two spacecrafts. The initial research on relative dynamics between two spacecrafts was investigated based on Newton’s second law [18]. This relative equation was followed and improved upon by Clohessy and Wiltshire [19], who proposed the relative dynamics model in a simple form known as the C-W equations when the leader spacecraft is deployed in a circular orbit. Afterwards, Brodsky and Shoham [20] proposed a translation and rotation coupling dynamic model based on a dual quaternion. In the space station rendezvous and docking mission, the same integrated dynamic model was used to guide and control the spacecraft [21]. The above research on relative dynamics focuses on rigid bodies. Specifically, the object of capture and control is a rigid body, and the assembly robot is also a rigid body. However, in order to save mission time and improve robot work efficiency, in on-orbit assembly missions, structure modules are often designed with large dimensions and a light weight [22]. This design results in structural modules exhibiting flexible characteristics. The rigid-flexible coupling effect must be taken into consideration during module transportation by rigid FFSR. Sun et al. and Zhang et al. [23–25] proposed the relative dynamics between two rigid-flexible coupling spacecrafts for an in-space assembly mission, which considered both the engineering application advantages of rotating reference and the compactness representation advantages of dual numbers. The current achievements focus on the relative dynamic modeling of rigid bodies or RFCSs with stable configuration.
Generally speaking, these two actions, configuration adjustment and module transportation, can be carried out separately. However, in order to improve the efficiency of the on-orbit assembly mission, achieving the two actions at the same time is a better choice, that is, integrated motion. The equations proposed in this paper are to describe the complex coupling dynamics of this integrated motion based on dual quaternion. The main contributions of this paper are summarized as follows: (a)The dual momentum and the dynamics model for RFCS with time-varying configuration were developed(b)The relative dynamics equations between two RFCSs with time-varying configuration were proposed
The organizational structure of this paper is as follows. The application scenarios, coordinate systems, and modeling assumptions were introduced in Section 2. Section 3 developed the six-dimensional spinor of the flexible modules motion, and the dynamic model of RFCS with time-varying configuration was established. Then, in Section 4, the relative dynamics model between two RFCSs with time-varying configuration was established, and the coupling effect of the on-orbit assembly was analyzed. In Section 5, the simulation results of the integrated model proposed in this paper are presented. Finally, Section 6 provides the conclusion of this article.
2. Application Scenarios, Coordinate Systems, and Modeling Assumptions
The application mission scenario was designed to be flexible modules postcapture operation and transportation by rigid FFSRs to a preassembled configuration. As shown in Figure 1, the RFCS consists of a flexible module and a FFSR; the flexible module and FFSR are connected by the capture device. In this paper, the capture device was assumed as a three-degree-of-freedom rotary pair; that is, the flexible module could perform a three-degree-of-freedom relative attitude motion relative to FFSR. In the on-orbit assembly mission, two actions were carried out at the same time for RFCS , that is, configuration adjustment action and module transportation action. The configuration adjustment action is to prepare the best pose for assembly through the three-degree-of-freedom relative rotation between the flexible module and the FFSR. The module transportation action is to transport the flexible modules to the desired locations near RFCS for integration. Figure 1 describes the in-space assembly mission and the coordinate systems. The following four coordinate systems are used in this paper: (a)Earth-centered inertial (ECI) coordinate system . The + and + axes of the ECI coordinate system point at the north pole of the Earth and the vernal equinox, respectively. + axis completes the right-hand set(b)Body-fixed (BF) coordinate system of RFCS . The origin of the BF coordinate system of RFCS locates at the connection point between the module and FFSR. The + and + axes point at the relative measuring sensor and the module, respectively. + axis completes the right-hand set(c)Body-fixed (BF) coordinate system of RFCS . The origin of the BF coordinate system of RFCS locates at the connection point between the module and FFSR. The + and + axes point at the relative measuring sensor and the module, respectively. + axis completes the right-hand set(d)Floating coordinate system of RFCS . The origin of coincides with the BF coordinate , and at the initial moment, the three-axis coordinate direction of coincides with the BF coordinate system . The difference is that the floating frame, , is a flexible module follow-up coordinate system
In this paper, at the initial moment, the origin and three-axis coordinate direction of the BF coordinate system of RFCS coincide with the floating frame of RFCS . This construction of the coordinate system can simplify the calculation of the dual momentum of RFCS with a time-varying configuration. Detailed calculations can be found in Section 3.2.
The integrated dynamic model of a rigid-flex combination system with time-varying configuration based on dual quaternion was derived under the following assumption.
Assumption 1. Compared to the size of the module and FFSR, the vibration displacement of the module is a small amount.
3. Dynamics of RFCS with Time-Varying Configuration
A dual quaternion is a dual number with each element of the dual a quaternion, that is , where and are both quaternions. is the dual unit defined as and . The spacecraft dynamic model expressed in dual quaternion can be expressed in the same mathematical framework, which makes the representation of the relative dynamic model compact. This model is generally known as the integrated dynamic model. The compactness of representation is positively correlated with the computational efficiency of the relative dynamic model.
3.1. Motion of Flexible Module
The finite-element principle is a method of turning a continuous infinite degree-of-freedom problem into a discrete finite degree-of-freedom problem. A more specific discussion on the finite-element principle can be found in Refs. [26, 27]. According to the finite-element principle, the velocity of an arbitrary discrete element of a flexible module relative to the COM of RFCS can be expressed as where is the velocity of with respect to the COM of RFCS . is the orbital velocity of the COM of RFCS b. is the angular velocity of RFCS . is the position vector from the COM of RFCS to the origin of the BF coordinate system . is the relative angular velocity between the flexible module and the FFSR. denotes the position vector from the origin of the BF coordinate system to . and denote the elastic position and elastic velocity of , respectively.
The dual velocity of the discrete element with respect to the COM of RFCS is
According to the algebra of dual numbers, the dual velocity of the discrete element with respect to the origin of , , in terms of components along the BF coordinate system is where
is the position vector from the origin of the BF coordinate system to the COM of RFCS in terms of components along the BF coordinate system . denotes the cross-product matrix of . The cross-product matrix for an arbitrary third-order vector is
The dual momentum of relative to the BF coordinate system is where and , , and denote the vectors , , and represented relative to the BF coordinate system , and , , and .
Substituting Equation (7) into Equation (6). According to Assumption 1, and . Therefore, the elastic displacement can be neglected in the calculation of Equation (6). Subsequently, Equation (6) can be further expressed as
Substituting Equation (1) and Equation (2) into Equation (8), the dual momentum of can be further expressed as
In Equation (9), the following operational relationship holds where .
According to Equations (10), (11), and (12), Equation (9) can be expressed in an explicit form
In Equation (13), , and the following operational relationship holds
According to the finite-element principle, the dual momentum of the flexible module with respect to the BF coordinate system can be expressed as
Substituting Equations (13), (14), (15), (16), and (17) into Equation (18), the dual momentum of the flexible module can be obtained.
In addition, let where denotes the moment of inertia of the flexible module. is the mass of the flexible module.
Having derived the discrete equation of the dual momentum, we perform the following model transformation on Equation (19).
As shown in Equation (21), the module vibration displacement can have a linear equation relationship with the modal coordinates, , through the matrix of eigenvectors of , . is the first derivative of .
According to Ref. [28], the following equation transformations can be obtained. where and are the rigid-flex translational and rotational coupling matrices, respectively. Substituting Equations (20), (21), and (22) into Equation (19), the dual momentum of the flexible module in hybrid coordinate can be expressed as
Representing Equation (16) in six-dimensional spinor form
By symbolizing Equation (24), we obtain where is the dual velocity of the flexible module relative to FFSR. The dual inertia matrix is a six-dimensional parameter matrix. The dual velocity is presented in terms of components along the BF coordinate system. is a new parameter for the dual representation of a rigid-flex combination system, which is defined as the rigid-flex dual coupling matrix. is also a new parameter for the dual representation of a rigid-flex combination system, which is defined as the dual-modal coordinates.
Taking the first derivative of Equation (25), we obtain
Equation (26) can be further expressed as
In addition, let where is the force between the FFSR and the flexible module. contains two parts, one is the driving force of FFSR acting on the flexible module, , and the other is the connection force between FFSR and the flexible module, .
3.2. Dual Number Representation of FFSR Motion
FFSR is the rigid part of RFCS , assuming is a nominal point of FFSR. The velocity of can be represented as
Note that is obtained relative to the COM of RFCS .
Furthermore, we can obtain the dual velocity of the particle . where . Similarly, is presented relative to the COM of RFCS .
Then, according to the operation rules of dual quaternion, the dual velocity relative to the origin of the BF coordinate system is obtained.
Note that is a dual vector in terms of components along the BF coordinate system.
Based on the dual velocity of , we can further obtain the dual momentum of .
Substituting Equation (32) into Equation (31), we perform specific operations on the dual momentum of and obtain
By simplifying Equation (33), one can obtain where
is the moment of inertia of FFSR. is the mass of FFSR.
Representing Equation (16) in six-dimensional spinor form, one can obtain
By symbolizing Equation (24), we obtain where is the dual inertia operator of FFSR.
Taking the first derivative of Equation (37), one can obtain
3.3. Dynamics Modeling of RFCS with Time-Varying Configuration
The dual momentum of RFCS is the sum of the dual momentum of the flexible module and rigid FFSR.
Equation (39) can be further expressed as
In addition, let where is the moment of inertia of RFCS . is the mass of RFCS .
In Equation (41), we can further obtain
Introducing Equation (42) into Equation (41), the dual inertia operator of RFCS can be expressed as
Substituting Equation (28) into Equation (40), the dynamics of the RFCS with time-varying configuration can be obtained.
In Equation (44), the first equation represents the six-degree-of-freedom dynamics of the whole system; the second equation represents the relative motion between the flexible module and the FFSR. Equations in Equation (44) are equations of the RFCS with time-varying configuration, which can also describe the 6 DOF motion of the spacecraft having rotational solar panels. The dynamic model of a satellite with rotational solar panels in Ref. [29] derived from the classical Newton-Euler method has the same form as Equation (44), which proves the correctness of the dynamic model proposed in this paper. Furthermore, Equation (44) is a general model which can describe the relative motion between two non-centroid points of spacecraft, while the classical Newton-Euler model cannot. In addition, the integrated relative dynamic equations of Equation (44) can improve the computational efficiency of the model.
In the rest of this section, an ideal physical scenario is assumed. In this scenario, the force exerted by the robot on the module, , is zero, and the module moves relative to the robot at a constant angular velocity, . Ignore motion damping and other disturbances between the robot and the module. Then, in this scenario, the dynamics of the RFCS with time-varying configuration can be expressed as
In Equation (45), the variable quantity is derived from . According to Refs. [30, 31], can be obtained through the following function: where is the natural frequency matrix of the flexible module, is the th natural frequency, denotes the damping ratio of the flexible module, and indicates the orbital velocity of RFCS under the action of external forces, excluding gravity.
4. Relative Dynamics between Two RFCS with Time-Varying Configuration
The relative motion state between RFCS and RFCS can be expressed as a dual quaternion. where denotes the relative dual quaternion state between two spacecrafts. ; denotes the relative position vector from to with respect to the body-fixed coordinate system of RFCS . denotes the relative attitude quaternion.
Furthermore, the relative kinematical equation can be represented as where ; . is the relative orbital velocity and is the relative angular velocity. is the relative dual velocity.
According to Ref. [21], can be derived from the following function: where and . is the orbital velocity of RFCS , is the angular velocity of RFCS , and the above two vectors are presented with respect to the body-fixed coordinate system of RFCS . is the dimensionality reduction operator. The dimensionality reduction operator for an arbitrary eight-dimensional vector is .
According to Equation (49), the following equation can be obtained.
Substituting Equation (44) into Equation (50), the relative dynamics between RFCS and RFCS is obtained.
In the ideal physical scenario assumed in Section 3.3, if the force exerted by the robot on the module, , is zero, the relative dynamics equation can further be represented as
5. Simulation Results
In this section, two sets of numerical simulations are provided. The first simulation was proposed to quantify the dynamics of RFCS with time-varying configuration in the ideal physical scenario assumed in Section 3.3 (Equation (45)). The influence of the relative attitude angular velocity, , on the six-degree-of-freedom motion of the RFCS was analyzed. The second simulation was provided to analyze the influence of the coordinate origin selection of the BF coordinate system on relative motion description.
The initial position vector and orbit velocity of RFCS with respect to the BF coordinate system are and .
The initial attitude of RFCS is set as and .
The dynamics equation (Equation (45)) is implemented in the case when the six-degree-of-freedom motion of RFCS is governed by gravity () and the gravity gradient moment (). The influence of other forces and noises are neglected. Table 1 lists the orbital parameters of RFCS , and Table 2 lists the characteristic parameters of RFCS .
5.1. Case 1: Effect of Relative Attitude Motion between the Flexible Module and FFSR on the Six-Degree-of-Freedom Motion of RFCS
Considering , three sets of relative attitude angular, , , and , were adopted, and the simulation results were obtained.
The blue lines, red lines, and green lines are the six-degree-of-freedom motion of RFCS when , , and , respectively. Figure 2 shows the orbit of RFCS. Figures 3 and 4 show the orbit velocity and three-axis position of RFCS, respectively. Figures 5 and 6 show the angular velocity and attitude quaternion of the RFCS. As shown in Figures 2–6, in three cases, the six-degree-of-freedom motion curves of RFCS basically coincide. It shows that under the ideal condition assumed in Section 3.3, the relative rotational angular velocity between the module and FFSR has little effect on the six-degree-of-freedom motion of RFCS. The simulation results are consistent with the physical reality and prove the validity of the mathematical model for RFCS with time-varying configuration.
5.2. Case 2: Effect Coordinate Origin Selection of on the Description of the Six-Degree-of-Freedom Motion of RFCS
Considering , three sets of coordinate origin selection, , , and , were adopted, and the simulation results were obtained:
In Figures 7–12, the blue lines represent the six-degree-of-freedom motion of RFCS when . The red lines represent the six-degree-of-freedom motion of RFCS when . The green lines represent the six-degree-of-freedom motion of RFCS when . The gray lines in Figures 8 and 9 represent the deviation curve between red and blue in Figure 7, and the black lines represent the deviation curve between green and red in Figure 7. As shown in Figures 11 and 12, the attitude angular velocity and attitude quaternion of RFCS do not change with the migration of the origin of the BF coordinate system. In Figures 7–10, as the distance between the origin of the BF coordinate system and the center of mass is larger, the position deviation of RFCS is larger. The positional deviation of the combined system reaches about 1 m when .
6. Conclusions
This paper proposed a dynamic model of rigid-flex combination systems with time-varying configuration. Dual quaternions were used to describe the six-dimensional spinor form of a rigid-flex combination system. The algebra of dual momentum for RFCS with time-varying configuration was developed, and the relative dynamics between two RFCSs with time-varying configuration for on-orbit assembly were provided. The simulation results show that the configuration changes have little effect on the six-degree-of-freedom motion of rigid-flex combination systems under the ideal physical scenario. The position error caused by the migration of the origin of the BF coordinate system is about 1 m. This shows that the selection of the origin of the BF coordinate system is key for the dynamic modeling of rigid-flex combination systems with time-varying configuration.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.